The question asks about the relationship between the ordinary (Pearson) correlation coefficient ($\rho_p$) and the Spearman rank correlation coefficient ($\rho_s$) when the Pearson correlation is exactly 1 for a given set of bivariate measurements.

Pearson Correlation Understanding

The Pearson correlation coefficient, $\rho_p$, measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to +1.

• $\rho_p = 1$: Indicates a perfect positive linear relationship. The data points lie exactly on a straight line with a positive slope.
• $\rho_p = -1$: Indicates a perfect negative linear relationship. The data points lie exactly on a straight line with a negative slope.
• $\rho_p = 0$: Indicates no linear relationship.

In this problem, we are given that $\rho_p = 1$. This means the bivariate measurements $(x_i, y_i)$ fall perfectly on a straight line with a positive slope. Since all $x_i$ are distinct and all $y_i$ are distinct, this perfect positive linear relationship implies that as $x$ increases, $y$ strictly increases.

Spearman Rank Correlation Understanding

The Spearman rank correlation coefficient, $\rho_s$, measures the strength and direction of a monotonic relationship between two variables. It is calculated by first ranking the data for each variable separately and then computing the Pearson correlation coefficient on these ranks. Its value also ranges from -1 to +1.

• $\rho_s = 1$: Indicates a perfect positive monotonic relationship. As one variable increases, the other consistently increases, but not necessarily in a linear fashion. The ranks of the two variables are perfectly positively correlated.
• $\rho_s = -1$: Indicates a perfect negative monotonic relationship. As one variable increases, the other consistently decreases. The ranks of the two variables are perfectly negatively correlated.
• $\rho_s = 0$: Indicates no monotonic relationship.

Relationship between Pearson and Spearman Correlation when $\rho_p = 1$

We are given $\rho_p = 1$. This means there is a perfect positive linear relationship between $x$ and $y$. If $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ lie on a straight line with a positive slope, say $y = a + bx$ with $b > 0$, and all $x_i$ are distinct, then if we order the $x$ values, the $y$ values will be ordered in the exact same way.

Let's consider the ranks:

• Assign ranks to $x_i$: $Rank(x_i)$
• Assign ranks to $y_i$: $Rank(y_i)$

Since $y = a + bx$ with $b > 0$, if $x_i < x_j$, then $y_i < y_j$. Because all $x_i$ and $y_i$ are distinct, a perfect positive linear relationship ensures that the order of the $x$ values is exactly the same as the order of the $y$ values.

For example, if $x_{(1)} < x_{(2)} < \dots < x_{(n)}$ are the ordered $x$ values, then $y_{(1)} < y_{(2)} < \dots < y_{(n)}$ must be the ordered $y$ values corresponding to the same data points. This means the observation with the smallest $x$ value also has the smallest $y$ value, the observation with the second smallest $x$ value also has the second smallest $y$ value, and so on.

Consequently, the rank of $x_i$ will be equal to the rank of $y_i$ for every data point $i$.

The Spearman rank correlation coefficient, $\rho_s$, is the Pearson correlation coefficient between the ranks $Rank(x_i)$ and $Rank(y_i)$. If $Rank(x_i) = Rank(y_i)$ for all $i$, then there is a perfect positive linear relationship between the ranks. The Pearson correlation calculated on perfectly linearly related data (in this case, the ranks which are identical) is 1.

Therefore, if $\rho_p = 1$, then $\rho_s = 1$.

Conclusion

Given that the Pearson correlation coefficient $\rho_p = 1$ for bivariate data where all $x_i$ and $y_i$ are distinct, this implies a perfect positive linear relationship. A perfect positive linear relationship is also a perfect positive monotonic relationship. The Spearman rank correlation coefficient measures the monotonic relationship between the ranks, and in the case of a perfect positive linear relationship, the ranks of $x$ and $y$ will be identical. The Pearson correlation between identical sets of ranks is 1. Hence, the Spearman rank correlation coefficient $\rho_s$ must be equal to 1.